How to model and animate a bifurcated path?
Curves cannot be forked to create branches. And if you create more than one segment within the same curve object you won't be able to adjust an animation of those segments independently. You need separate objects.

As an example:

Let's say you animate the bevel Factor of a curve by animating the End, setting a keyframe with value 0 at frame 0, and one with a value of 100 at frame 60.

To make keyframes just hover the cursor over the value and press I.



Then you create a new segment with a new curve object and align it to start where you want the fork to start. A single curve cannot be animated at different rates in different segments, that is why it has to be a separate new curve. In this example I'm using the snap to function (Shift+S) to place the new segments precisely over the older curve.



Animate the start and end of the bevel for the second curve so that it starts at the point you want.